Trigonometric Identities & Equations – Lesson

Trigonometric identities are mathematical statements that are true for all values of the angles in a right triangle. They relate the ratios of the sides of a right triangle to the angles formed by those sides. Some of the most common trigonometric identities include:

1. Pythagorean Identity: cos²θ + sin²θ = 1, where θ is an angle in a right triangle.
2. Reciprocal Identities: cotθ = 1/tanθ, secθ = 1/cosθ, and cscθ = 1/sinθ, where θ is an angle in a right triangle.
3. Quotient Identities: tanθ = sinθ/cosθ, and cotθ = cosθ/sinθ, where θ is an angle in a right triangle.
4. Cofunction Identities: sin(90° – θ) = cosθ, and cos(90° – θ) = sinθ, where θ is an angle in a right triangle.
5. Sum-to-Product Identities: sin(α + β) = sinαcosβ + cosαsinβ, and sin(α – β) = sinαcosβ – cosαsinβ, where α and β are angles in a right triangle.
6. Difference-to-Product Identities: cos(α – β) = cosαcosβ + sinαsinβ, and cos(α + β) = cosαcosβ – sinαsinβ, where α and β are angles in a right triangle.

These identities are fundamental to trigonometry and are used to simplify expressions, solve problems, and understand the behavior of periodic functions.

Trigonometric double angle identities are mathematical statements that relate the trigonometric functions of double angles to the trigonometric functions of half angles. These identities are useful for simplifying and solving problems involving trigonometry. Some of the most common trigonometric double angle identities include:

1. Sin(2θ) = 2sinθcosθ
2. Cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ
3. Tan(2θ) = (2tanθ)/(1 – tan²θ)
4. Sin²θ = (1 – cos(2θ))/2
5. Cos²θ = (1 + cos(2θ))/2
6. Sin(θ/2) = ±√((1 – cosθ)/2)
7. Cos(θ/2) = ±√((1 + cosθ)/2)

These identities are derived using the sum-to-product and difference-to-product identities and can be used to simplify and solve problems involving trigonometry. They are also used in the study of complex numbers, differential equations, and other areas of mathematics.